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# Hamiltonian algebroids

# 0.1 Introduction

*Hamiltonian algebroids* are generalizations of the Lie algebras of canonical transformations, but cannot be considered just a special case of Lie algebroids. They are instead a special case of a quantum algebroid.

###### Definition 0.1.

Let $X$ and $Y$ be two vector fields on a smooth manifold $M$, represented here as operators acting on functions. Their commutator, or Lie bracket, $L$, is :

$\displaystyle[X,Y](f)=X(Y(f))-Y(X(f)).$ |

Moreover, consider the classical configuration space $Q=\mathbb{R}^{3}$ of a classical, mechanical system, or particle whose phase space is the cotangent bundle $T^{*}\mathbb{R}^{3}\cong\mathbb{R}^{6}$, for which the space of (classical)
observables is taken to be the real vector space of smooth functions on $M$, and with T being an element
of a Jordan-Lie (Poisson) algebra whose definition is also recalled next. Thus, one defines as in classical dynamics the *Poisson algebra* as a Jordan algebra in which $\circ$ is associative. We recall that one needs to consider first a specific algebra (defined as a vector space $E$ over a ground field (typically $\mathbb{R}$ or $\mathbb{C}$)) equipped with a bilinear and distributive multiplication $\circ$ . Then one defines a *Jordan algebra* (over $\mathbb{R}$), as a a specific algebra over $\mathbb{R}$ for which:

$\begin{aligned}\displaystyle S\circ T&\displaystyle=T\circ S~{},\\ \displaystyle S\circ(T\circ S^{2})&\displaystyle=(S\circ T)\circ S^{2},\end{% aligned},$

for all elements $S,T$ of this algebra.

Then, the usual algebraic types of morphisms automorphism, isomorphism, etc.) apply to a
Jordan-Lie (Poisson) algebra defined as a real vector space $U_{{\mathbb{R}}}$ together with a *Jordan product* $\circ$ and *Poisson bracket*

$\{~{},~{}\}$, satisfying :

- 1.
for all $S,T\in U_{{\mathbb{R}}},$

$\begin{aligned}\displaystyle S\circ T&\displaystyle=T\circ S\\ \displaystyle\{S,T\}&\displaystyle=-\{T,S\}\end{aligned}$

- 2.
the

*Leibniz rule*holds$\{S,T\circ W\}=\{S,T\}\circ W+T\circ\{S,W\}$ for all $S,T,W\in U_{{\mathbb{R}}}$, along with

- 3.
the

*Jacobi identity*:$\{S,\{T,W\}\}=\{\{S,T\},W\}+\{T,\{S,W\}\}$ - 4.
for some $\hslash^{2}\in\mathbb{R}$, there is the

*associator identity*:$(S\circ T)\circ W-S\circ(T\circ W)=\frac{1}{4}\hslash^{2}\{\{S,W\},T\}~{}.$

Thus, the canonical transformations of the Poisson sigma model phase space specified by the Jordan-Lie (Poisson) algebra (also Poisson algebra), which is determined by both the Poisson bracket and the *Jordan product* $\circ$, define a *Hamiltonian algebroid* with the Lie brackets $L$ related to such a Poisson structure on the target space.

## Mathematics Subject Classification

81P05*no label found*81R15

*no label found*81R10

*no label found*81R05

*no label found*81R50

*no label found*

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